1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

In a linear space, 0 times an element of the space need not be 0?

  1. Feb 11, 2013 #1
    1. The problem statement, all variables and given/known data
    Hello, we are starting to get to Banach spaces and thus linear normed spaces in a functional analysis class and I am realizing that I don't have much experience or intuition with these spaces. So I was reading over the requirements for a linear space in my notes and was surprised that there was not a property that 0[itex]\cdot[/itex]x = 0. Is this just implicitly assumed to be true, or is this really not a property of a linear space?

    To be more precise, I have an intuitive understanding of what a linear space means if we are considering Euclidean vectors, but if it is just some abstract space that follows the rules of a linear space, then I don't really know what it means to multiply by a scalar. For example, I know what the output of multiplying a vector by a scalar will be but in a more abstract setting it doesn't seem like such a rule needs to be given, only the property that the result will still be in the space. So I am trying to not make the mistake of applying what I know about normal Euclidean vectors to more general concepts.


    2. Relevant equations
    The properties of a linear space


    3. The attempt at a solution
     
    Last edited: Feb 11, 2013
  2. jcsd
  3. Feb 11, 2013 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, if x is any vector then 0x=0 (where the first 0 is the zero scalar and the second is the zero vector). That's a property of ALL vector spaces and a Banach space is a type of vector space. So they may not have felt a need to specifically mention it.
     
  4. Feb 11, 2013 #3
    So vector space is interchangeable with linear space?
     
  5. Feb 11, 2013 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I would think so.
     
  6. Feb 11, 2013 #5

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The property ##0x = 0## does not need to be listed as part of the definition, as it can be derived from the distributivity property: a vector space must satisfy ##(a+b)x = ax + bx## for all scalars ##a## and ##b## and all vectors ##x##. Choosing ##a=b=0##, this implies that ##(0+0)x = 0x + 0x##. As ##0 + 0 = 0## is true in any field by definition of the additive identity ##0##, the left hand side simplifies to ##0x##, and we have ##0x = 0x + 0x##. Subtracting ##0x## from both sides, we get ##0 = 0x##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: In a linear space, 0 times an element of the space need not be 0?
Loading...